When is a given set of PDEs part of an elliptic system?

Abstract

We investigate the following question: Given a set of $k$ partial differential equations for $m$ unknowns, where $k < m$, can we find $m-k$ additional equations in such a way that the full set of equations forms an elliptic system? We formulate a maximal rank condition which is obviously necessary. In general, however, the maximal rank condition is sufficient only if we allow the introduction of additional variables, not just additional equations. In particular, the equation ${\rm div}\,u=0$ can be part of an elliptic system for the components of the vector field $u$ only if the space dimension is 1,2,4, or 8.